2. Let us define the following functions f(x, y) = x² + 2y² y² g(x, y) = x² + (a) (3 points) (*) Find the Oxf, ayf, and (Vg)(x, y). (b) (3 points) (*) Find a critical point of g(x, y) and determine whether it is a local maximum, a local minimum, or a saddle. (c) (3 points) (*) Draw the level set of g for level 9. (d) (3 points) (*) Note that g(3 cos(t), 9 sin(t)) = 9 for any t = [0,2π]. Find t that maximises f(3 cos(t), 9 sin(t)). (e) (4 points) (**) Using the Lagrange multiplier method, find (x, y) that maximise f(x, y) subject to g(x, y) = 9.
2. Let us define the following functions f(x, y) = x² + 2y² y² g(x, y) = x² + (a) (3 points) (*) Find the Oxf, ayf, and (Vg)(x, y). (b) (3 points) (*) Find a critical point of g(x, y) and determine whether it is a local maximum, a local minimum, or a saddle. (c) (3 points) (*) Draw the level set of g for level 9. (d) (3 points) (*) Note that g(3 cos(t), 9 sin(t)) = 9 for any t = [0,2π]. Find t that maximises f(3 cos(t), 9 sin(t)). (e) (4 points) (**) Using the Lagrange multiplier method, find (x, y) that maximise f(x, y) subject to g(x, y) = 9.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Need help with part e). Please explain each step and neatly type up. Thank you :)
![2. Let us define the following functions
f(x, y) = x² + 2y²
y²
9
g(x, y) = x² +
(a) (3 points) (*) Find the Oxf, ayf, and (Vg)(x, y).
(b) (3 points) (*) Find a critical point of g(x, y) and determine whether it is a local
maximum, a local minimum, or a saddle.
(c) (3 points) (*) Draw the level set of g for level 9.
(d) (3 points) (*) Note that g(3 cos(t), 9 sin(t)) = 9 for any t = [0,2π]. Find t that
maximises f(3 cos(t), 9 sin(t)).
(e) (4 points) (**) Using the Lagrange multiplier method, find (x, y) that maximise
f(x, y) subject to g(x, y) = 9.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6fe1fc35-672a-49fd-831b-9642c77888ed%2F29b3e3f6-c433-48fb-914c-f47038732640%2F9p75c3h_processed.png&w=3840&q=75)
Transcribed Image Text:2. Let us define the following functions
f(x, y) = x² + 2y²
y²
9
g(x, y) = x² +
(a) (3 points) (*) Find the Oxf, ayf, and (Vg)(x, y).
(b) (3 points) (*) Find a critical point of g(x, y) and determine whether it is a local
maximum, a local minimum, or a saddle.
(c) (3 points) (*) Draw the level set of g for level 9.
(d) (3 points) (*) Note that g(3 cos(t), 9 sin(t)) = 9 for any t = [0,2π]. Find t that
maximises f(3 cos(t), 9 sin(t)).
(e) (4 points) (**) Using the Lagrange multiplier method, find (x, y) that maximise
f(x, y) subject to g(x, y) = 9.
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